Rain is falling vertically with a speed of $35 \; m s^{-1}$. $A$ woman rides a bicycle with a speed of $12 \; m s^{-1}$ in the east to west direction. What is the direction in which she should hold her umbrella?

(N/A) Let $v_r$ represent the velocity of rain and $v_b$ represent the velocity of the bicycle,both with respect to the ground.
Since the woman is riding a bicycle,the velocity of rain as experienced by her is the velocity of rain relative to the bicycle,given by:
$v_{rb} = v_r - v_b$
This relative velocity vector $v_{rb}$ makes an angle $\theta$ with the vertical,as shown in the figure.
It is given by:
$\tan \theta = \frac{|v_b|}{|v_r|} = \frac{12}{35} \approx 0.343$
$\theta = \tan^{-1}(0.343) \approx 19^{\circ}$
Therefore,the woman should hold her umbrella at an angle of about $19^{\circ}$ with the vertical towards the west.